Some thoughts in no particular order:
Community charge - varies from LA to LA, but if they're both in hospital for an extended period you may be able to get community charge waved.
Insurance: most policies have a clause which says insurance doesn't apply if home is unoccupied for more than 60 days continuously. So you need to tell the insurance companies the situation. They may insist you move to a more expensive "unoccupied premises" insurance, but that's a better option than something drastic happening and finding that insurance won't cover it.
Since we're in January, you'll need to either maintain the house at a reasonable temperature (insurance companies specify 15deg) so the pipes don't freeze, or turn the CH and the water off completely.
Other people have a responsibility to ensure your parents aren't being the victims of fraud, and most companies are very good at not altering any financial arrangements at the behest of a stranger on the phone. LAs are sometimes more lax, so you may find it OK to deal with community charge, but you'll need to contact Office of Public Guardian/Court of Protection/Age UK for how to deal with the rest. Or try the hospital social worker. They encounter this question a lot.
Your mother being assessed for dementia is not a reason for not being able to get PoA - all that's required is she should have capacity to understand the implications of what she's doing on the day that she signs the document that gives you PoA. It sounds like your father doesn't have that capacity at the moment. But PoA won't help you in the short term - it takes several weeks to get it registered.
Don't feel bad about writing cheques for him - you're safeguarding him in allowing to continue to manage his affairs. I used to cope by writing cheques and letters for my father and just getting him to sign them (If you think of it, this is exactly how the Prime Minister works - civil servants write his letters, and he just appends his signature), but I can't see how to make this work in hospital, because somebody else would have to explain to him, and this may not work, even if anybody had the time to do it!
I believe that if they go into an assessment unit they don't pay for 6 weeks. Up to 6 weeks. As soon as it's decided that a) rehabilitation has been successful or b) the the person will need long term care, the entitlement stops.
They stop being responsible for their care home fees once their individual savings (including share of the house) drop to below £23000. There's an income limit too. There will be plenty of elderly people without family to deal with finances, so Social Services should be able to advise you (though remember the loyalty of a Social Services finance person will always be towards getting money for the LA rather than ensuring your parents' financial best interests).